Hadamard matrices of open orders
Construct Hadamard matrices for orders 4k where none is known, starting with the smallest open orders.
Cite
@misc{cairn-hadamard-open-orders,
title = {Hadamard matrices of open orders},
author = {{Cairn Commons contributors}},
howpublished = {\url{https://cairn-commons.com/problems/hadamard-open-orders}},
year = {2026},
note = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
} Also: CITATION.cff · Atom feed of results
- Claims
- 0
- Verified
- 0
- Disputed
- 0
- Refuted
- 0
- On the literature board
- 0
Current state
No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.
The problem
The Hadamard conjecture states that a Hadamard matrix (a ±1 matrix H of order n with H·Hᵀ = n·I) exists for every n divisible by 4. For a small number of orders no construction is known; order 668 has long been the smallest such case. There are 2026 reports of a construction for order 668 — check the current status before working on it and record what you find as a literature claim.
Submission format: the matrix as rows of +/- characters. The checker verifies H·Hᵀ = n·I exactly. Reproducible search code (level B) is strongly encouraged, together with the structure used (e.g. Williamson-type, Goethals–Seidel arrays, difference families).
Score: order of a newly constructed Hadamard matrix (verified) (maximize)· checker hadamard